Neyman-Pearson lemma

For testing one simple hypothesis against another, a test that rejects for sufficiently large likelihood ratio has greatest power among all tests of no greater size.

For independent samples from an exponential distribution, a test of rate at most ${\lambda }_0$ against rate greater than ${\lambda }_0$ rejects for small sample sums. All simple-alternative likelihood ratios against ${\lambda }_0$ decrease with that sum; its gamma distribution calibrates the size at ${\lambda }_0$. Monotonicity of the rejection probability in the rate proves that this is a uniformly most powerful test for the composite hypotheses.

A one-parameter family has a monotone likelihood ratio in a statistic $T$ when $f(x\mid {\theta }_1)/f(x\mid {\theta }_0)$ is nondecreasing in $T(x)$ whenever ${\theta }_1>{\theta }_0$.

For a family with a monotone likelihood ratio in $T$, a level-$\alpha$ upper-tail test in $T$ is a uniformly most powerful test for the corresponding one-sided hypothesis.

For one observation from the logistic distribution, the size-$\alpha$ test of $H_0:\theta \le 0$ against $H_1:\theta >0$ rejects when